Asymptotic tracking with prescribed transient behaviour for linear systems
The problem of asymptotic tracking of reference signals is considered in the context of m-input, m-output linear systems (A,B,C) with the following structural properties: (i) CB is sign definite (but not necessarily symmetric), (ii) the zero dynamics are exponentially stable. The class of reference signals is the set of all possible solutions of a fixed, stable, linear, homogeneous differential equation (with associated characteristic polynomial α). The first control objective is asymptotic tracking, by the system output y = Cx, of any reference signal . The second objective is guaranteed error e = y − r transient performance: e should evolve within a prescribed performance funnel (determined by a function φ). Both objectives are achieved simultaneously by an internal model in series with a proportional time-varying error feedback t↦ u(t)=−ν (k(t)) e(t), where ν is a smooth function with the properties and , and k(t) is generated via a nonlinear function of the product ‖ e(t)‖ φ (t) The feedback structure essentially exploits an intrinsic high-gain property of the system by ensuring that, if (t,e(t)) approaches the funnel boundary, then the gain attains values sufficiently large to preclude boundary contact.